Analytical representations of solutions to multidimensional inverse problems for evolutionary equations (Q1386564)

The paper deals with the many-dimensional inverse problem for the evolutionary equation: find the complex-valued functions \(w(x,t)\), \(\lambda(x)\), and \(\mu(x)\), \(x\in\mathbb{R}^n\), \(a\leq t\leq b\), such that \[ {\partial^2 w\over \partial t^2}+ Aw= \lambda(x)f(t) +\mu (x) \varphi(t), \] \[ w\left|_{t=a} =w_a(x),\quad {\partial w\over \partial t}\right |_{t=a} =w_a'(x), \quad w\left|_{t=b} =w_b(x),\quad {\partial w\over \partial t} \right|_{t=b} =w_b'(x). \] Here, \(A\) is a linear operator that acts with respect to the variables \(x=(x_1,x_2, \dots,x_n)\) of the real Euclidean space \(\mathbb{R}^n\) and has the symbol \(\widetilde A(\xi)\), \(\xi\in \mathbb{R}^n\): \(\widetilde A(\xi)\in C^\infty (\mathbb{R}^n)\), \(Ae^{ix\xi} =e^{ix\xi} \widetilde A(\xi)\).